Abstract. This chapter outlines the research, development and perspectives of quantum neural networks -a burgeoning new field which integrates classical neurocomputing with quantum computation [1]. It is argued that the study of quantum neural networks may give us both new undestanding of brain function as well as unprecedented possibilities in creating new systems for information processing, including solving classically intractable problems, associative memory with exponential capacity and possibly overcoming the limitations posed by the Church-Turing thesis. Keywords. Quantum neural networks, associative memory, entanglement, many universes interpretation Why quantum neural networks?There are two main reasons to discuss quantum neural networks. One has its origin in arguments for the essential role which quantum processes play in the living brain. For example, Roger Penrose has argued that a new physics binding quantum phenomena with general relativity can explain such mental abilities as understanding, awareness and consciousness [2]. However, this approach advocates the study of intracellular structures, such as microtubules rather than that of the networks of neurons themselves [3]. A second motivation is the possibility that the field of classical artificial neural networks can be generalized to the quantum domain by eclectic combination of that field with the promising new field of quantum computing [4]. Both considerations suggest new understanding of mind and brain function as well as new unprecedented abilities in information processing. Here we consider quantum neural networks as the next natural step in the evolution of neurocomputing systems, focusing our attention on artificial rather than biological systems. We outline different approaches to the

This paper discusses a model of quantum associative memory which generalizes the completing associative memory proposed by Ventura and Martinez. Similar to this model, our system is based on Grover's well known algorithm for searching an unsorted quantum database. However, the model presented in this paper suggests the use of a distributed query of general form. It is demonstrated that spurious memories form an unavoidable part of the quantum associative memory model; however, the very presence of these spurious states provides the possibility of organizing a controlled process of data retrieval using a specially formed initial state of the quantum database and also of the transformation performed upon it. Concrete examples illustrating the properties of the proposed model are also presented.

A lattice-gas algorithm for the one-dimensional diffusion equation is realized using radio frequency pulses in a one-dimensional spin system. The model is a large array of quantum two-qubit nodes interconnected by the nearest-neighbor classical communication channels. We present a quantum protocol for implementation of the quantum collision operator and a method for initialization and reinitialization of quantum states. Numerical simulations of the quantum-classical dynamics are in good agreement with the analytic solution for the diffusion equation.

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We present a multiagent model illustrating the emergence of two different quantum statistics, Bose-Einstein and Fermi-Dirac, in a friendly population of individuals with the right-brain dominance and in a competitive population of individuals with the left-brain hemisphere dominance, correspondingly. Doing so, we adduce the arguments that Lefebvre's "algebra of conscience" can be used in a natural way to describe decision-making strategies of agents simulating people with different brain dominance. One can suggest that the emergence of the two principal statistical distributions is able to illustrate different types of society organization and also to be used in order to simulate market phenomena and psychic disorders, when a switching of hemisphere dominance is involved.